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circle. Suppose we wish to go from one point A to another B on
the surface of the Earth, keeping always on the surface in
passing from A to B, and suppose further that we wish to make
the journey by the shortest way possible. This is the problem of
'great circle sailing'. Imagine a plane passed through A, B, and
the centre of the Earth (there is one, and only one, such plane).
This plane cuts the surface in a great circle. To make our
shortest journey we go from A to B along the shorter of the two
arcs of this great circle joining them. If A9 B happen to lie at
the extremities of a diameter, we may go by either arc.
The preceding example introduces, an important definition,
that of a geodesic on a surface, which will now be explained. It
has just been seen that the shortest distance joining two points
on a sphere, the distance itself being measured on the surface,
is an arc of the great circle joining them. \Ye have also seen that
the longest distance joining the two points is the other arc of the
same great circle, except in the case when the points are ends of
a diameter, when shortest and longest are equal. In the chapter
on Fermat 'greatest* and 'least* were subsumed under the
common name "extreme*, or * extremum'. We recall now one
usual definition of a straight-line segment joining two points in
a plane - 'the shortest distance between two points'. Trans-
ferring this to the sphere, we say that to straight line in the
plane corresponds great circle on the sphere. Since the Greek
word for the Earth is the first syllable ge (yfj) of geodesic we
call all extrema joining any two points on any surface the
geodesies of that surface. Thus in a plane the geodesies are
Euclid's straight lines; on a sphere they are great circles. A
geodesic can be visualized as the position taken by a string
stretched as tight as possible between two points on a surface.
Now, in navigation at least, an ocean is not thought of as a
flat surface (Euclidean plane) if even moderate distances are
concerned; it is taken for what it very approximately is, namely,
a part of the surface of a sphere, and the geometry of great
circle sailing is not Euclid's. Thus Euclid's is not the only
geometry of human utility. On the plane two geodesies inter-
sect in exactly one point unless they are parallel, when they do
not intersect (in Euclidean geometry); but on the sphere any